"Social Work" is not a once-size-fits-all degree. The are bachelor level social work degrees, masters level social work degrees, and masters degrees with post-graduate training. Some masters level social workers have ACSW after their name -- meaning they are "academy of social work certified". Additionally trained masters level social workers can not only screen patients, they can provide the therapy in some clinics.

Should social works screen patients? MA level, sure. Sorting out the merely unhappy from the seriously depressed isn't all that hard--the janitor could probably do that. It's the more complex things like identifying borderline personality disorder, schizoid affective disorder, bipolar, schizophrenic, and such diagnoses that require higher level diagnostic training.

Social workers and doctorate-level therapists can't prescribe medicine, generally, but are qualified to provide therapy and support for mental patients. Psychiatrists normally do not actually provide much therapy--it's a rare practice that does. Mostly they write prescriptions, because medicine is often the main ingredient in effective therapy. All the good counseling in the world won't calm down a psychotic individual.